IIT JEE , AIEEE Forum - Mathematics , Algebra

donnowatnametochoose

If A³ = 0 then I + A + A² =

arpan1

I + A + A² = I

RATE ME

joyfrancis

yup..a zero matrix satisfys A^3=0 ..so

I + (zero matrix) + (zero matrix)^2 = I ..:)

ashish_banga

if A^3 =0 it need not necessarily imply A is a null matrix

donnowatnametochoose

Pl elaborate:)

This is what i did

Let I+A+A²=k =>A+A²=Ak (Multiply by A)

=>I+A=k(Multiply by A^-1) the answer given is (I-A)^-1

donnowatnametochoose

@ashish :Agreed

arpan1

nothing like that

let y =I + A + A²

Multiply A on both sides

Ay = AI + A²

Again multiplying A on both sides, we get

A²y = A²I

HENCE , A = I

HOPE U GOT IT

joyfrancis

i know yaar...but for objectives you can do it this way...

donnowatnametochoose

@arpan1: Don't think......you cannot cancel matrices!Evaluate y.Whats wrong in the method given,anyways both answers are wrong.

donnowatnametochoose

The answer given is (I-A)^-1

arpan1

HEY I DID NOT CANCEL THE TERMS, JUST TOOK A^-1 TWICE

donnowatnametochoose

Please find the mistake in my method and the answer given is not I.

ramkumar_november

use binomial theorem for any index .....

$(I-A)^{-1}=I+A+\frac{(-1).(-2)}{2!}.A^2+\frac{(-1).(-2).(-3)}{3!}.A^3+......... till\;\;\infty$

since A³ = 0 .... all terms get cancelled except I + A +A² .....

so $I+A+A^2=(I-A)^{-1}$

donnowatnametochoose

@ramkumar:Whats wrong in this:

Let I+A+A²=k =>A+A²=Ak (Multiply by A)

=>I+A=k(Multiply by A^-1)..you went from the answer?Anyways good thinking.